Calculations of NMR shielding constants have been carried out at the HF, MP2, B3LYP and KT3 levels of theory using the Gaussian096and DALTON20117 program packages. In all cases the GIAO technique was utilised to ensure gauge independence. Since this study is concerned with convergence of shielding constants with basis set size, the molec-
ular geometries employed are not particularly significant. All calculations have been carried out at the HF/cc-pVDZ equilibrium geometry. The geometries of all molecules studied are presented in Tables A.16-A.83. No vibrational or thermal averaging of the shielding constants has been undertaken, although this would be necessary for compar- ison with experimental values.
4.3.1 Basis Set Convergence
The rate of convergence of MP2 shielding constants with the number of basis functions was examined for several common types of basis set: Jensen’s pcS-n and aug-pcS-n
bases with n=0-4;55 Dunning’s cc-pVxZ,47,48 aug-cc-pVxZ50 and cc-pCVxZ51 bases withx=D, T, Q, 5; the Pople style bases STO-3G, 6-31G** and 6-311G(2df,2pd);43–46 and Ahlrich’s SVP,52TZV53and QZVP54bases. Basis sets not featured in the standard implementation for each program package were obtained from EMSL.151Given the high
computational cost of these MP2 calculations, shielding constants were evaluated for all atoms in a test set of relatively small molecules: NH3, H2O, HF, CH4, C2H4, C2H2, HCN,
CH3F, F2, CO and N2. Aside from ensuring tractability, these molecules were chosen for
possessing lone pairs and/or multiple bonds (methane being the sole exception), and thus representing cases for which correlation might be important. Given that Jensen’s pcS-n
and aug-pcS-n basis sets have been shown to provide rapid convergence for shielding constants at DFT levels, the largest basis set (aug-pcS-4) has been taken to represent the basis set limit. Hence, all shielding constants are presented relative to the MP2/aug- pcS-4 value.
4.3.2 Locally Dense Basis Sets
All calculations with locally dense basis sets have been carried out using the HF, B3LYP and KT3 methods, and the pcS-n basis sets. When investigating basis set partitionings we distinguish between through bond and through space interactions. Through bond effects are those that occur between atoms that are close in the sense of connectivity, namely they are separated by only a few bonds. Through space interactions are those that occur between atoms that are in close spatial proximity, but well separated in terms of connectivity. The distinction was largely practical, as through bond effects can be investigated for relatively small and simple molecules, while through space effects require molecules that are large enough to have groups that are well separated in terms
of connectivity. For this reason we group delocalised π-systems together with through space interactions, even though this is a through bond effect, since it requires larger systems to investigate.
When investigating through bond effects, calculations were performed on the fol- lowing molecules: 2-methoxyethylamine, 1,2-diaminoethane, methylethylamine, diethy- lamine, 1,3-diaminopropane, 3-hydroxy-1-aminopropane, azetidine, pyrollidine, n-propyl- amine, cyclopropylamine, isopropylamine, cyclobutylamine, propylene imine, allylamine, n-Butylamine, n-propanol, i-propanol, n-butanol, i-butanol, s-butanol, t-butanol, 2- chloroethanol, isobutylamine, ethandiol, 2-aminoethanol, allyl alcohol, sec-butylamine and tert-butylamine. Coordinates for this test set can be found in Tables A.27-A.54. These molecules are within a size range of 4-5 heavy atoms, and are thus small enough that uniform pcS-4 calculations are feasible, though large enough to contain functional groups with α, β, and more distant substituent groups. None of these molecules have appreciable through space interactions, and they thus represent an appropriate test set for examining through bond effects. There are a total of 244 hydrogen and 126 heavy atoms in these molecules.
For the purpose of investigating through space effects, molecules with some form of long range interactions (ie. explicit charges, intramolecular hydrogen bonds and delo- calised systems) were selected. Calculations were performed on the following molecules: 2-aminopropanol, aniline, anisole, benzene, butanal, butanone, ethanoic acid (depro- tonated), ethoxybenzene, glycine (neutral and zwitterion), isopropanoic acid (depro- tonated), 2-cyanoethanol, malondialdehyde, n-butylamine (protonated), n-propylamine (protonated), n-methylaniline, nitrobenzene, propanal, propanamide, propanoic acid (protonated and deprotonated), sec-butylamine (protonated), 3-aminopropanol, isobuty- lamine (protonated), isopropylamine (protonated), acetylacetone, acrolein, acrylic acid and allylcyanide. The coordinates for these molecules can be found in Tables A.55-A.83. These molecules are significantly larger than those of the first test set, with the largest containing 9 heavy atoms. For this reason the investigation was limited to the HF level of theory. There are a total of 209 hydrogens and 160 heavy atoms in these molecules.
For both test sets the largest basis set (pcS-4) has been taken to represent the conver- gence limit, with all shielding constants presented as deviations from the corresponding HF, B3LYP and KT3 values obtained with the pcS-4 basis set on all atoms in the molecule.
The initial partitioning schemes in this work were based on connectivity rather than distance. This allowed the effect of each atom and group to be studied systematically. Additional distance and group based partitionings were considered to improve the shield- ings for particular long range interactions.
Atoms are considered connected if the distance between them is less than the sum of their covalent radii (plus a small tolerance of 0.4˚A). This criteria was chosen to reproduce the ordinary chemical assignment of bonds. Two different schemes to partition the basis set throughout the molecule were considered.
(i) The atom-based partition: An atom is chosen for which the shielding constant is calculated. This is henceforth referred to as the “focus” atom. The basis set for this atom is denoted pcSx. All atoms bonded to the focus atom are assigned a common basis, denoted pcSy. All other atoms in the molecule are assigned a common basis, denoted pcSz. The complete basis set is then denoted as pcSx/pcSy/pcSz.
(ii) The group based partition: The bonded atoms are assigned to groups containing a single heavy atom and any hydrogens bonded to it. A group is chosen for which the shielding constants are calculated. This will henceforth be referred to as the “focus” group. The common basis set for each atom in this group is denoted pcSx. All atoms in groups that are bonded to the focus group are assigned a common basis, denoted pcSy. All other atoms in the molecule are assigned a common basis, denoted pcSz. The complete basis set is then denoted as pcSx/pcSy/pcSzG.
Other group based partitions are considered, including carboxylic acids, aromatic rings, nitro groups and amides. These are described as they are implemented. They are, however, problematic in that they contain several heavy atoms, all of which are assigned a pcSx basis set when part of the “focus” group. For this reason, an alternative method of grouping, referred to as soft-grouping, is considered wherever possible for these functional groups. Under this scheme, if the “focus” group is in a soft-group, then the atoms in the focus group are assigned a pcSx basis set, while the remainder of the soft-group is assigned a pcSy basis set.
In addition, distance based criteria are specified in which all atoms with a given distance of any atom in the focus group is given a pcSy basis set.
A partitioning scheme is locally dense if: